characterize separation capacity

This study addresses the problem of quantifying the feature separation capacity of convolutional neural networks in classification tasks. Building upon Cover’s function counting theory and integrating tools from combinatorics with an analysis of scattering network architectures, we extend the classical theoretical framework to derive, for the first time, a practical expression for the separation capacity specific to scattering networks. Our analysis elucidates the critical influence of architectural components—such as the number of filters and network depth—on separation capability. Furthermore, this work establishes the first theoretically grounded design principles for scattering networks, offering both significant theoretical insight and practical utility for network architecture development.

This work investigates the design of pooling-free scattering networks employing fixed monomial nonlinearities to maximize separability for data with low intrinsic dimensionality. By integrating frame theory, geometric measure theory, and moment analysis, the study provides the first geometric characterization of a scattering network’s separation capacity, establishing theoretical bounds for feature extractors operating on low-dimensional rectifiable data. The core contribution consists of two practical design principles: the network’s filters must span a sufficiently broad frequency range, and the frame formed by these filters—when coupled with the data’s geometric structure through a coupling matrix—must exhibit a well-conditioned condition number. These criteria jointly ensure significantly enhanced separation performance, offering concrete guidance for the construction of effective scattering architectures tailored to geometrically structured low-dimensional data.

This work addresses the limitation of existing neural classifiers that rely on linear readouts and struggle to capture the geometric structure of class representations, particularly under few-shot conditions where unilateral affine separability cannot be properly assessed. The authors propose a directional Linear Separability Metric (LSM) that quantifies the minimal proportion of competing-class samples intruding into an affine half-space containing all samples of a target class. LSM exhibits asymmetry, class-level granularity, target normalization, and invariance under full-rank linear transformations, thereby distinguishing the effects of linear reparameterizations from those of information loss or nonlinear distortions. An efficient penalty-based affine search algorithm is introduced to estimate LSM in high-dimensional feature spaces while preserving the original discrete constraints. Experiments demonstrate that LSM effectively reveals class intrusion phenomena induced by components such as coordinate gating, offering a novel tool for analyzing the geometry of neural representations.

Existing classification complexity measures and clustering validity indices (CVIs) inadequately balance inter-cluster separation and intra-cluster connectivity in density-based clustering, leading to insufficient assessment of whether classes constitute meaningful density clusters. To address this, we propose the Density-based Clustering Separation–Connectivity Index (DCSI), the first unified metric jointly modeling both properties: inter-cluster separation is quantified via density-reachability, while intra-cluster cohesion integrates path-based connectivity within clusters and local neighborhood geometry. DCSI effectively identifies touching or overlapping yet density-inseparable clusters—resolving key limitations of conventional CVIs in evaluating density-based algorithms such as DBSCAN. Experiments demonstrate that DCSI achieves high correlation with the Adjusted Rand Index (ARI) on synthetic benchmarks and accurately detects overlapping structures unsuitable for hard density clustering on real-world data. The index combines theoretical rigor with practical utility for density clustering validation.

Subspace clustering in high-dimensional data often yields multiple semantically distinct subspaces, yet existing methods require manual specification of both the number of subspaces and the number of clusters within each—rendering them parameter-sensitive and poorly interpretable. This paper proposes an automatic, non-redundant multi-subspace clustering framework. First, it introduces the Minimum Description Length (MDL) principle to non-redundant clustering, enabling joint, adaptive inference of both the optimal number of subspaces and the cluster count per subspace. Second, it designs a split-merge-based greedy search strategy coupled with a subspace-level outlier encoding mechanism, allowing simultaneous outlier detection. Evaluated on multiple benchmark datasets, the method achieves competitive accuracy against state-of-the-art approaches while significantly improving parameter robustness, model interpretability, and practical applicability.

研究了高维空间中快速度量分解算法，针对ℓ∞和ℓ2空间提出新算法，提高分解效率和参数性能。

该研究通过消除几何学框架探讨局部最优对象能否由共享部署规则实现，分析信息、架构等因素对缺陷修复的影响。

This study addresses the challenge of quantifying population segregation by proposing a general measurement framework based on convex functions over the probability simplex. Methodologically, it leverages Jensen’s inequality to define segregation as the discrepancy between local and global diversity, thereby constructing Jensen-information-based measures. To incorporate spatial structure, three strategies are introduced: spatial smoothing, aggregation, and local Jensen information. By integrating convex optimization with spatial statistics theory, this work establishes a comprehensive mathematical system for measuring segregation. Computational examples validate the effectiveness of various spatial modeling approaches. Ultimately, this research provides a unified theoretical foundation for quantifying the interplay between population attributes and spatial separation.

该研究使用Gromov的horofunction紧化方法，解决在非局部紧空间中进行鲁棒性分析的问题，特别是在无限维空间中的应用。

This study addresses the data separation phenomenon in categorical response models, which frequently renders maximum likelihood estimates nonexistent or non-unique, thereby severely compromising the reliability of statistical inference. To resolve this issue, we develop divoRce, an R package grounded in structural vector theory that integrates computational geometry, linear programming, and rational arithmetic solvers. Supporting multiple link functions and third-party extensions, this toolkit enables existence testing, type classification, and identification of the variables responsible for separation. This work contributes the most comprehensive exact diagnostic suite currently available in the literature, encompassing virtually all categorical models. We recommend incorporating divoRce into standard analytical workflows to ensure modeling robustness and safeguard the validity of downstream inferential procedures.